Given $ \overrightarrow{OA}\perp\overrightarrow{OC}$, $ m \angle AOB = 5x + 31$, and $ m \angle BOC = 8x + 46$, find $m\angle BOC$. $O$ $A$ $C$ $B$
From the diagram, we see that together ${\angle AOB}$ and ${\angle BOC}$ form ${\angle AOC}$ , so $ {m\angle AOB} + {m\angle BOC} = {m\angle AOC}$ Since we are given that $\overrightarrow{OA}\perp\overrightarrow{OC}$ , we know ${m\angle AOC = 90}$ Substitute in the expressions that were given for each measure: $ {5x + 31} + {8x + 46} = {90}$ Combine like terms: $ 13x + 77 = 90$ Subtract $77$ from both sides: $ 13x = 13$ Divide both sides by $13$ to find $x$ $ x = 1$ Substitute $1$ for $x$ in the expression that was given for $m\angle BOC$ $ m\angle BOC = 8({1}) + 46$ Simplify: $ {m\angle BOC = 8 + 46}$ So ${m\angle BOC = 54}$.